The function f(x) = tanx - x

The correct option is:
is an increasing function on $[0,\frac{\pi }{2})$

To determine whether the function $f\left(x\right)=\tan x-x$ is increasing or decreasing, we can analyze the sign of its first derivative with respect to $x$.

Recall that a function is strictly increasing on an interval if its derivative is strictly positive for all points in that interval.
Let us find the derivative of $f\left(x\right)$:
$$f^{\prime }\left(x\right)=\frac{d}{dx}(\tan x-x)$$
Using the standard differentiation rules, we have:
$$f^{\prime }\left(x\right)={\sec }^{2}x-1$$

Using the fundamental trigonometric identity ${\sec }^{2}x-1={\tan }^{2}x$, we can rewrite the derivative as:
$$f^{\prime }\left(x\right)={\tan }^{2}x$$

Now, let us evaluate the sign of $f^{\prime }\left(x\right)$ on the given interval $[0,\frac{\pi }{2})$:
1. For $x=0$, we have $f^{\prime }\left(0\right)={\tan }^2\left(0\right)=0$.
2. For all $x\in (0,\frac{\pi }{2})$, the value of $\tan x$ is positive, which means that its square is strictly positive:
$$f^{\prime }\left(x\right)={\tan }^{2}x>0$$

Since $f^{\prime }\left(x\right)\ge 0$ on $[0,\frac{\pi }{2})$ and the derivative is zero only at a single isolated point ( $x=0$), the function $f\left(x\right)$ is strictly increasing on the interval $[0,\frac{\pi }{2})$.